$1.(34$ marks$)$ A company makes two types of tractors. After deducting all direct costs, each

Model $1$ tractor gives a profit of $$3400,$ while each Model $2$ tractor gives a profit of $$1200.$ Every

tractor goes through three assembly$-$lines. Every Model $1$ tractor takes $100$ minutes in Assembly$-$line

$1,180$ minutes in Assembly$-$line $2,$ and $50$ minutes in Assembly$-$line $3.$ Every Model $2$ tractor takes

$80$ minutes in Assembly$-$line $1,90$ minutes in Assembly$-$line $2,$ and $150$ minutes in Assembly$-$line

$3.$ During the next planning period, Assembly$-$lines $1,2,$ and $3$ will be open $1600,2160,$ and $2000$

hours respectively. There is a contractual obligation to produce at least $250$ Model $1$ tractors during

the planning period. They want the production of Model $1$ tractors to be no more than $80\%$ of the

total production.

$($a$)(10$ marks$)$ Formulate a linear optimization model for this situation.

$2$